Until recently I was under the impression that Catholics are unanimous in their beliefs and don't have the kind of fragmentation that Protestants have.

But I’ve learned that there are several small groups/institutions within the main church.

There are traditionalist catholics that reject reforms of Vatican council.
There are breakaway churches that maintain same doctrines but do not submit to authority of Rome. Then there are terms like Conclavist Catholics and Sedevacantists (I have no idea what it means I'm not knowledgeable on the subject).

My question is - In the 2000 years of history what significant factions have existed within the Catholic church? How have they been different from each other?

When you ask about "breakaway churches", etc, you are talking about a very different thing than "institutions within the main church". There's a big difference between a sub-group created for administrative purposes and a dissident group. Like, to the best of my knowledge the Jesuits and the Franciscans are not opposing the pope or any official church teaching. They're more like the Catholic equivalent of a corporation creating a new department or a task force.
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JayJul 5 '12 at 4:47

@Jay "There's a big difference between a sub-group created for administrative purposes and a dissident group". Agreed. But I'm completely in the dark on both of these, so answers on either are welcome. I'm just trying to survey the catholic religious landscape.
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Monika MichaelJul 5 '12 at 15:33

3 Answers
3

There is one Church. There are hundreds of factions (after all, any organization of sufficient size will have political infighting). Then there are many people who claim to be Catholic but are really heretics in sheep's clothing.

Sedevacantists are people who deny the validity of the current Pope as well as the Second Vatican Council. That is more-or-less like denying Nicea. You can't do it and manage to escape the title "heretic."

Conclavists are (most often) sedevacatists who have gone so far as to say, "since VAII was heresy, the bishops who supported it are heretics. Heretics are incapable of properly electing a pope. Therefore, there is no pope and the See is vacant."

I do think that it is a good idea to expand a bit on your title, even though that is not your full question. There are quite a few different liturgical and spiritual groups inside the Catholic Church. Most commonly, you will find Byzantine and other Eastern groups who closely resemble the Orthodox in both practice and theological expressions, but there are other, smaller sub-organizations in the Church (such as the Ambrosian Rite (which is largely liturgical) and the Anglican Ordinariate (which is only a few years old and closely resembles the ways of the Anglican Church)). Depending how you subdivide it, you could even argue that every major order (or even every abbey) can be considered a subset with its own customs, beliefs, and traditions. For that matter, even within a diocese, you will find different parishes behave different ways.

And now, a joke:

Why is it easier to convert an atheist than a conclavist?At least an atheist believes that there's a Pope.

Looove the joke.. :-) But what about opus dei and jesuits. What is their relation with the church?
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Monika MichaelJul 4 '12 at 18:48

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Jesuits are a religious order, like the Franciscans, Dominicans, or Benedictines. Individual members of any of these groups can become heretics, but the groups are all under the See of Rome. Opus Dei is something called a "Personal Prelature" meaning it has its own group of bishops and is not restricted to a diocese. They are relatively autonomous (and enjoy some freedom religious orders don't), but they are still under the Pope.
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Ignatius TheophorusJul 4 '12 at 20:41

Okay. Thanks for pointing out all those names of institutions and orders. Please share some more if you know. That way I know what and where to look to do further research.
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Monika MichaelJul 4 '12 at 21:03

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Isn’t sedevacantist the term for those who believe that the see is vacant? Without proper sources, your statement Sedevacantists are people who deny the Second Vatican Council seems incorrect to me.
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törzsmókusOct 4 '13 at 4:20

@törzsmókus: In general, yes. But today, sedevacantists often use the Second Vatican Council as proof that that popes from John XXIII onwards are really anti-popes.
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GeremiaMay 7 '14 at 5:42

'Old Catholics' reject Vatican I, the SSPX reject certain elements of Vatican II, SSPV reject all of Vatican II but reject the Thuc line of episcopal consecration, CMRI accept the Thuc line and reject Vatican II, all of the above believe in Baptism of Desire and Baptism of Blood. Conclavists reject the Pope and have elected their own. Sedevacantists assert there is no Pope because the man claiming to be a Pope has removed himself from the Church automatically by promoting heresy.

Modern 'Catholics' accept Vatican II, but they have to reject most of the older teachings, like for example, Separation of Church and State was condemned by Pius IX but its now promoted, the idea of non-Catholic Saints was unanimously condemned before Vatican II, but now its explicitly promoted. The conversion of the Jews was promoted before Vatican II, but now its rejected. That's a few examples of hundreds.

All of these groups would say that they are the real Catholics, and that the other groups are heretics or schismatics.

There are 23 different types of Catholic Christians. What most call Roman Catholic are actually Latin Rite Catholic while the Eastern Orthodox are made up of the rest of the 22. I'm what you call Latin Rite but the Eastern Rites are older.